Absolutely, including the redeemed.Everyone has a sin nature.
The difference between those believers and those unbelievers is that believers also have a new nature which battles against the nature of the flesh.
Almost.That is where the preaching of the gospel comes in. The gospel itself has the power unto salvation. Faith comes from hearing the gospel preached. The gospel alone draws us to God.
God draws, and uses the Scriptures to instill faith.
Again, the preaching is the power of God unto salvation, but the draw to hear and to understand resides with God who gives the ears to understand and changes the core (heart) to accept. This is presented (as shown in another post) in Hebrews restatement of the promise given through the prophet Ezekiel.The church takes the gospel to the lost world who does not seek God. There the preaching of the gospel has the power unto salvation.
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In another place it states that not all have ears to hear, but that those who do are to listen with understanding and results.